Pavel Exner and Andrea Mantile
On the optimization of the principal eigenvalue for single-centre point-interaction operators in a bounded region
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ABSTRACT. We investigate relations between spectral properties of a single-centre
point-interaction Hamiltonian describing a particle confined to a bounded domain $\Omega\subset\mathbb{R}^{d},\: d=2,3$, with Dirichlet boundary, and the geometry of $\Omega$. For this class of operators Krein's formula yields an explicit representation of the resolvent in terms of the integral kernel of the unperturbed one, $\left( -\Delta_{\Omega}^{D}+z\right) ^{-1}$. We use a moving plane analysis to characterize the behaviour of the ground-state energy of the Hamiltonian with respect to the point-interaction position and the shape of $\Omega$, in particular, we establish some conditions showing how to place the interaction to optimize the principal eigenvalue.